Question

Describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$ \theta=\frac{\pi}{4} $$

Short answer

The equation describes a line at a \( 45^\circ \) angle with the x-axis.

Step-by-step solution

Understand the Polar Equation

The given polar equation is \( \theta = \frac{\pi}{4} \). In polar coordinates, \( \theta \) represents the angle from the positive x-axis. Hence, this equation describes a line where every point on the line forms an angle \( \frac{\pi}{4} \) with the positive x-axis.

Convert Polar Equation to Rectangular Coordinates

To convert from polar to rectangular coordinates, we use the relationships: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Since \( \theta = \frac{\pi}{4} \), we have that \( y/x = \tan(\frac{\pi}{4}) \). Simplifying, \( \tan(\frac{\pi}{4}) = 1 \), so \( y = x \). This is the rectangular form of the line.

Confirm the Description

The rectangular equation \( y = x \) confirms that the graph is a straight line making a \( 45^\circ \) or \( \frac{\pi}{4} \) angle with the x-axis, consistent with the polar description. In both the polar and rectangular forms, we have the same line that passes through the origin with equal x and y coordinates for any point on the line.

Key concepts

Rectangular Coordinates

In the world of trigonometry and graphs, we often come across different coordinate systems. One of the most common systems is the rectangular coordinate system. It uses two axes, X and Y, to pinpoint locations on a plane. Imagine this system as a grid with horizontal (x-axis) and vertical (y-axis) lines intersecting each other at right angles at the origin. In this grid, any point can be represented by a pair of numbers ( x, y), where **x** shows how far along the horizontal axis the point is, and **y** shows how far along the vertical axis it is. This system is also known as the Cartesian coordinate system, a staple in algebra and geometry.

• **Rectangular coordinates** are perfect for linear equations, straight lines, and right-angle discussions.
• They help in understanding the relationship between variables more clearly, as they split the movement into two straightforward directions (horizontal and vertical).

Recognizing these coordinates is crucial for translating other systems like polar coordinates into a familiar framework that we can easily calculate with.

Angle Conversion

Understanding angles is key to moving between different coordinate systems, especially from polar to rectangular. In the context of converting polar coordinates where the angle is given (denoted by \( \theta \)), angle conversion becomes important.For the polar equation \( \theta = \frac{\pi}{4} \), this refers to an angle that is measured counterclockwise from the positive x-axis. This angle is equivalent to \( 45^{\circ} \) in degrees. Converting radians to degrees or vice versa involves using the relationships \( \pi \, ext{radians} = 180^{\circ} \).

• Angles in polar coordinates describe direction, while rectanglar coordinates use them to find opposition between x and y.
• It aids in transitioning a path from radial and angular terms to linear distance terms in rectangular coordinates.

For conversions, use consistent calculations to ensure accuracy. Recognizing common angles such as \( \frac{\pi}{4}, \frac{\pi}{2} \), etc., in both radians and degrees is a useful skill.

Graphing Equations

Graphing mathematical equations is a way of visualizing the relationships between variables. For polar equations like \( \theta = \frac{\pi}{4} \), the graph describes a specific direction rather than a specific radius or distance from the origin.When this equation is converted into a rectangular equation, it takes the familiar form \( y = x \). Visualizing this in the rectangular coordinate system, we see a straight line that forms a 45-degree angle with the x-axis. This angle indicates the direction of the slope of the line, meaning for every one unit you move to the right, you move one unit up, thus keeping the line straight through the origin.

• Graphing helps see how changes in one variable affect another, either in polar or rectangular coordinates.
• Provides a clear visual representation, especially with lines where every point follows consistent rules - like equal x and y in the case of \( y = x \).

Through graphing, abstract mathematical concepts take a more concrete form, allowing better understanding and insight into the equation's behavior.

Coordinate Systems

There are two primary coordinate systems we discuss: polar and rectangular. Each has its strengths and typical use cases.

• Rectangular (Cartesian) System: Ideal for problems involving linear equations and measuring distances along axes. Here points are represented as \( (x, y) \).
• Polar System: Best for situations where you’re dealing with angles and radial distances from a central point. Points are given as \( (r, \theta) \).

In the polar system, angles and distances paint a picture of the position rather than coordinates in a static grid. It’s very useful for circular or spiral patterns, while linear equations are easily represented in the rectangular system.Changing from polar to rectangular coordinates or vice versa requires specific formulas like \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Each system has its unique advantages, and understanding conversions between them allows for flexible problem-solving capabilities.